Question

1. What is (f⋅g)(x)?

f(x)=x^2+2x−6

g(x)=x^4+3

Enter your answer in standard form in the box.

2. Question
WHAT IS : (f−g)(x)? its not the same question as 1

f(x)=x^4−x^2+9

g(x)=x^3+3x^2+12

Enter your answer in standard form in the box.

Answer 1

`\textsf{1.} \quad \left(f \cdot g \right)(x)=x^6+2x^5 -6x^4+3x^2+6x -18`

`\textsf{2.} \quad\left(f -g \right)(x)=- x^4+x^2+2x-9`

Explanation:

To calculate (f ⋅ g)(x), multiply function f(x) by function g(x) by distributing the parentheses:

`\begin{aligned}\left(f \cdot g \right)(x)&=(x^2+2x-6)(x^4+3)\\&=x^2 \cdot x^4 +x^2 \cdot 3+2x \cdot x^4+2x \cdot 3 -6 \cdot x^4 -6 \cdot 3\\&=x^6+3x^2+2x^5+6x -6x^4 -18\\&=x^6+2x^5 -6x^4+3x^2+6x -18\\\end{aligned}`

To calculate (f - g)(x), subtract function g(x) from function f(x) by applying the distributive law -(a + b) = -a - b:

`\begin{aligned}\left(f -g \right)(x)&=(x^2+2x-6) - (x^4+3)\\&=x^2+2x-6 - x^4-3\\&=- x^4+x^2+2x-6 -3\\&=- x^4+x^2+2x-9\end{aligned}`

Note: The standard form of a polynomial is expressed by writing the term with the highest degree first, followed by the other terms in order of decreasing powers.